Cubic Graphs and Graphical Solutions

Drawing a cubic from a table

A cubic function has the form $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$ and $a, b, c, d \in \mathbb{Z}$. At this level you draw it the same way as a quadratic: a table of values, plot the points, join them with a smooth curve.

The graph has one of two basic shapes, depending on the sign of $a$:

• If $a > 0$, the curve starts low on the left and finishes high on the right.

• If $a < 0$, the curve starts high on the left and finishes low on the right.

The reason is that for large $|x|$ the $ax^3$ term is far bigger than the others. It is negative for large negative $x$ and positive for large positive $x$ (and the other way round if $a < 0$).

A cubic can have two turning points (a local maximum and a local minimum), or none. Try the sliders below and watch how the curve changes.

The table above gives the points on this graph. The turning points are marked.

Intercepts and roots

The graph crosses the $y$-axis where $x = 0$, so the $y$-intercept is $f(0)$. For a cubic this is always the constant term $d$.

The graph crosses the $x$-axis where $f(x) = 0$. These $x$-values are the roots of $f(x) = 0$. A cubic has at most three roots, and at least one.

To find roots from a graph, read the $x$-values at the points where the curve meets the $x$-axis.

If you only have a table of values, look for a change of sign. If $f(a)$ and $f(b)$ have opposite signs, the graph must cross the $x$-axis somewhere between $a$ and $b$, because a cubic graph has no breaks.

If you can factorise the cubic, the roots follow from the factors. For example, $x^3 - 4x = x(x-2)(x+2)$ has roots $0$, $2$ and $-2$.

Solving f(x) = k and f(x) = g(x) graphically

The equation $f(x) = 0$ asks where the graph meets the $x$-axis. You can use the same idea for other equations:

• $f(x) = k$: draw the horizontal line $y = k$ on the same axes. The $x$-values where it meets the curve are the solutions.

• $f(x) = g(x)$: draw both graphs on the same axes. The $x$-values of the points where they meet are the solutions.

Graphical solutions are estimates, so read them to the accuracy the scale allows (for example, to one decimal place).

Check by algebra: $x^3 - 3x = x$ gives $x^3 - 4x = 0$, so $x(x-2)(x+2) = 0$, which agrees.

You can also turn $f(x) = g(x)$ into a single equation: $f(x) = g(x)$ is the same as $f(x) - g(x) = 0$. For $x^3 - 3x = x + 1$ this is $x^3 - 4x - 1 = 0$. Whichever form you use, the solutions are the same numbers.

Turning points and sketching

A sketch shows the important features without plotting every point: the intercepts, the turning points and the overall shape.

To find the turning points use the derivative:

1. Find $f'(x)$ and solve $f'(x) = 0$.

2. Find $f(x)$ at each of these $x$-values to get the $y$-coordinates.

3. Use $f''(x)$: if $f''(x) > 0$ it is a local minimum, and if $f''(x) < 0$ it is a local maximum.

This is the graph in the first figure. Now we can close the opening question. The line $y = k$ meets the curve three times exactly when $k$ lies strictly between the local minimum value and the local maximum value, here when $-2 < k < 2$. In the figure, $y = 1$ lies in that range and cuts the curve three times. If $k = 2$ or $k = -2$ the line touches at the turning point and there are two solutions. Outside that range there is only one. That answers the road question: the road passes through a height exactly three times when that height lies between the heights of the dip and the crest.

To sketch from factors:

Finding the equation from points

If a graph passes through a point $(p, q)$, then $f(p) = q$. Put the coordinates into $f(x) = x^3 + bx^2 + cx + d$ to get an equation in the unknown coefficients. Use as many points as there are unknowns, then solve the equations.

A point with $x = 0$ gives $d$ straight away, which saves work.

Once you know a root, say $x = 1$, show it by substitution: $f(1) = 0$. Then $(x - 1)$ is a factor (factor theorem), and you divide it out to get a quadratic factor, whose roots are the other roots. A cubic may have one, two or three real roots, so always check the quadratic factor with the quadratic formula (it may give no real solutions).

Summary

• A cubic $f(x) = ax^3 + bx^2 + cx + d$ is drawn from a table of values. If $a > 0$ it goes from low-left to high-right, and if $a < 0$ from high-left to low-right.

• The $y$-intercept is $f(0) = d$. The $x$-intercepts are the roots of $f(x) = 0$. A sign change between two values of $x$ means a root lies between them.

• To solve $f(x) = k$, draw $y = k$. To solve $f(x) = g(x)$, draw both graphs. Read the $x$-values of the points where they meet.

• Turning points: solve $f'(x) = 0$, then use $f''(x)$: positive means a local minimum, negative means a local maximum. $f(x) = k$ has three solutions when $k$ is strictly between the two turning point values.

• To find coefficients, substitute each point into $f(x)$ and solve the equations. Check every point afterwards.